Problem: Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({\cos(\pi) + i \sin(\pi)}) ^ {19} $
Solution: Let's express our complex number in Euler form first. $ {\cos(\pi) + i \sin(\pi)} = { e^{\pi i}} $ Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{\pi i}}) ^ {19} = e ^ {19 \cdot (\pi i)} $ The angle of the result is $19 \cdot \pi$ , which is $19\pi$ $19\pi$ is more than $2 \pi$ . It is a common practice to keep complex number angles between $0$ and $2 \pi$ , because $e^{2 \pi i} = (e^{\pi i}) ^ 2 = (-1) ^ 2 = 1$ . We will now subtract the nearest multiple of $2 \pi$ from the angle. $ 19\pi - 18\pi = \pi $ Our result is $ e^{\pi i}$. Converting this back from Euler form, we get $\cos(\pi) + i \sin(\pi)$.